Learn Spanish (Latin America): ¿Eres tranquilo o inseguro?

You only change o to u when the word immediately after it starts with an "o" or "ho" sound (for example, malo u oscuro). Since inseguro starts with an "i" sound, the normal o stays exactly as it is.

Why do we use eres (from ser) instead of estás (from estar)? Aren't feelings temporary?

In this sentence, the speaker is asking about your general personality traits: "Are you a calm or insecure type of person?" If you wanted to ask how someone is feeling right at this moment, you would use estar: ¿Estás tranquilo? (Are you feeling calm right now?).

How would this sentence change if I were asking a woman?

You would need to change both adjectives to their feminine forms: ¿Eres tranquila o insegura?. Because both words describe the person you are talking to, they must match that person's gender.

I've heard people say ¡Tranquilo! to mean "Don't worry." Is that the same word?

Yes! As an adjective, tranquilo means "calm" or "peaceful." But Spanish speakers also use it as a common phrase (¡Tranquilo! or ¡Tranquila!) to tell someone to calm down, relax, or not to worry about a situation.